On a Gauss-Givental Representation of Quantum Toda Chain Wave Function

We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U(\mathfrak{gl}(N))$ in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter $Q$-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.